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14    LOCALLY CONVEX SPACES       67

(12.14.11) Let E, E' be two locally convex spaces. Let Y be a directed set of
seminorms defining the topology 0/E, and Tf a set of seminorms defining the
topology ofE'. Let u : E -> E' be a linear mapping. Then the following conditions
are equivalent:

(a)    u is continuous;

(b)   for each seminorm p' e F', there exists a neighborhood of 0 in E on
which the function x\-+p'(u(x)) is bounded (or, equivalently (12.14.2), the semi-
norm p' o u on E is continuous)',

(c)   for each seminorm p' e P, there exists a seminorm p e F and a real
number c> 0 such thatp'(u(x)) <£ c -p(x)for allxeE.

From the definition of the neighborhoods of 0 in E and E', condition (c)
implies that u is continuous at the point 0, hence everywhere (12.8.4). So
(c) implies (a). The condition (a) implies that the function p' ° u is continuous
on E (because p' is continuous on E'), hence (b). Finally, suppose that (b) is
satisfied. Then by hypothesis there exist two numbers r > 0 and a > 0 and a
seminorm p e F such that the relation p(z) <j r implies p'(u(z)) <| a. Let x be
any point of E and let A be a scalar strictly greater than 0 such that p(Xx) =
Ap(x) g r. Then p'(u(kx)) g a, so that p'(u(x)) <; #/! If /?(jc) = 0, we can
take 1 to be arbitrarily large, and therefore p'(u(x)) = 0; if XX) > 0> we ta^e
A = r/p(x)9 and then we have p'(u(x)) <i (0/r) • XX)- So in either case the
condition (c) is satisfied, with c = a/r.                                               Q.E.D.

(12.14.11.1)   Likewise, if u is a multilinear mapping of a product

E! x E2 x - • • x Er

of locally convex spaces into a locally convex space E', the continuity of u
implies that for each seminorm p' e F' there exist neighborhoods Vy of 0 in
E/ (1 ^y ^ «) such that the function (xl9 ..., #B)'-»Jp'(«(*i, • - - , *„)) is
bounded in V{ x • • • x Vn. If Fy is a directed set of seminorms defining the
topology of Ey (1 £j£ri)9 this latter condition implies that there exist
numbers a > 0, r7 > 0 (1 ^j ^n) and seminorms /^ 6 Fy (1 ^y :g /z) such that
the relations pj(zj)£rj(l£J£ri) imply p'(u(zl9 ...,%„))£ a. Now let
(#!,..., ^) be any point of E! x • • • x En , and for each j let Xj > 0 be such
that /?y(Ay Xj) = Ajpfaj) ^ ry; then it follows that

We deduce as above that ifpj(xj) = 0 for some index y, then lj may be chosen
arbitrarily large, and hence p (u(xly . . . , xj) = 0. If on the other hand Pj(xj) ^ 0
for all y, we may take Aj = rj/pfaj) for 1 gy g «; putting c = a/rA . . . rB,ological vector spaces.